The power generated by an electrical circuit (in watts) as a function of its current $c$ (in amperes) is modeled by $P(x)=-15x(x-8)$ What is the maximum power possible?
Explanation: The circuit's power is modeled by a quadratic function, whose graph is a parabola. The maximum power is reached at the vertex. So in order to find the maximum power, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $P(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} P(x)&=0 \\\\ -15x(x-8)&=0 \\\\ \swarrow &\searrow \\\\ -15x=0\text{ or }&x-8=0 \\\\ x={0}\text{ or }&x={8} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({8})}{2}=\dfrac82= 4$ The vertex's $x$ -coordinate is $ 4$. Now let's find $P({4})$ : $\begin{aligned} P( 4)&=-15( 4)( 4-8) \\\\ &=-15(4)(-4) \\\\ &=240 \end{aligned}$ In conclusion, the maximum power is $240$ watts.